Achromatic metasurface optical components by dispersive phase compensation

ABSTRACT

Multi-wavelength light is directed to an optic including a substrate and achromatic metasurface optical components deposited on a surface of the substrate. The achromatic metasurface optical components comprise a pattern of dielectric resonators. The dielectric resonators have nonperiodic gap distances between adjacent dielectric resonators; and each dielectric resonator has a width, w, that is distinct from the width of other dielectric resonators. A plurality of wavelengths of interest selected from the wavelengths of the multi-wavelength light are deflected with the achromatic metasurface optical components at a shared angle or to or from a focal point at a shared focal length.

RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.15/534,642, which is the National Stage of International Application No.PCT/US2015/064930, filed 10 Dec. 2015, the entire contents of which areincorporated herein by reference.

This application claims the benefit of U.S. Provisional Application No.62/090,172, filed 10 Dec. 2014, the entire content of which isincorporated herein by reference.

GOVERNMENT SUPPORT

This invention was made with government support under Grant No.FA9550-12-1-0289 awarded by the Air Force Office of Scientific Research.The Government has certain rights in the invention.

BACKGROUND

Refractive and diffractive optical components share many similaritieswhen they are used with monochromatic light. If we illuminate a prismand a grating with a laser beam, they will both bend the incoming light.In a similar fashion, a spherical lens and a diffractive lens (zoneplate) both focus light. However, the behavior of refractive optics anddiffractive optics is very different when they are used to manipulatebroadband light. A prism with normal dispersion will deflect the longerwavelengths to a smaller angle compared to the shorter wavelengths; adiffraction grating, instead, does the opposite. Likewise, the focaldistance for a refractive lens in the visible wavelengths will be largerfor red light than for blue, while the contrary occurs for a diffractivelens.

This contrasting behavior arises because two different principles areused to shape the light. Wavefront control in refractive optics isobtained by gradual phase accumulation as the light propagates through amaterial of a given refractive index, n=n(λ), on account of materialdispersion. In most transparent materials, the refractive indexdecreases with increasing wavelength (“normal dispersion”) over thevisible region. Since the deflection angle, θ, of a prism increases withthe index, n(λ) while a lens focal length, f, is inversely proportionalto n(λ), the resulting effect of refractive optics 11 is the one shownin images A and B of FIG. 1 .

A diffractive optical element (DOE) 13, instead, operates by means ofinterference of light transmitted through an amplitude or phase mask.The beam deflection angle and the focal length, respectively, aredirectly and inversely proportional to A (images C and D of FIG. 1 ),generating a dispersion opposite to that of standard refractive devices.Although for many applications a spatial separation of differentwavelengths is desirable (spectrometers, monochromators, wavelengthdivision multiplexing (WDM)), in many others this spatial separationrepresents a problem. For example, the dependence of the focal distanceon A produces chromatic aberrations and is responsible for thedegradation of the quality of an imaging system. We note that thewavelength dependence is typically much more pronounced in diffractiveoptics than in refractive optics. Materials used to make high-qualityrefractive optics can have very low dispersion; and in some cases,materials with opposite dispersion are used to cancel out the effect(e.g., achromatic doublets).

Another difference between these technologies is represented by theefficiency with which a desired function is achieved. In refractiveoptics, the efficiency can be very high and is limited only by materiallosses, fabrication imperfections, and interface reflections. Indiffractive optics, instead, the presence of higher diffraction ordersimposes severe limitations on performance. On the other hand,diffractive optical elements have the advantage of being relativelyflat, light and often low cost. Blazed gratings and Fresnel lenses arediffractive optical devices with an analog phase profile, and thus theyare simultaneously refractive and diffractive. As such, they integratesome benefits of both technologies (e.g., small footprint and highefficiency); but they still suffer from strong chromatic aberrations.Multi-order diffractive (MOD) lenses overcome this limitation by usingthicker phase profiles optimized such that the phase differencecorresponds to an integer number of 2π for each wavelength. With thisapproach, one can in principle obtain a set of wavelengths that arechromatically corrected (I). The realization of thick, analog phaseprofiles, however, is challenging for conventional technologies, such asgreyscale lithography or diamond turning.

Metasurfaces are thin optical components that rely on a differentapproach for light control; a dense arrangement of subwavelengthresonators is designed to modify the optical response of the interface.As shown previously [N. Yu, et al., “Light Propagation with PhaseDiscontinuities: Generalized Laws of Reflection and Refraction,” 334Science 333-37 (2011), and PCT Patent Application Publication No. WO2013/033591 A1], the resonant nature of the scatterers introduces alocal abrupt phase shift in the incident wavefront making it possible tomold the scattered light at will and enabling a new class of planarphotonics components (i.e., flat optics) [see N. Yu, et al., “Flatoptics: Controlling wavefronts with optical antenna metasurfaces,” IEEEJ. Sel. Top. Quantum Electron. 19(3), 4700423 (2013), and N. Yu, F.Capasso, “Flat optics with designer metasurfaces,” 13 Nat. Materials139-150 (2014). Different types of resonators (metallic or dielectricantennas, apertures in metallic films, etc.) have been used todemonstrate various flat optical devices, including blazed gratings,lenses, holograms, polarizers, and wave plates. The metasurface approachis unique in that it provides continuous control of the phase profile(i.e., from 0 to 27c) with a binary structure (only two levels ofthickness). Metasurfaces also circumvent the fundamental limitation ofmultiple diffraction orders typical of binary diffractive optics whilesimultaneously maintaining the size, weight, and ease-of-fabricationadvantages compared to refractive optics.

Metasurface-based optical devices demonstrated so far, however, areaffected by large chromatic aberrations (i.e., strongwavelength-dependence). Research efforts have recently shown thatrelatively “broadband” optical metasurfaces can be achieved. The claimof large bandwidth usually refers to the broadband response of theresonators, which is the result of the high radiation losses necessaryfor high scattering efficiency and, to a lesser extent, of theabsorption losses. As a consequence, the phase function implemented bythe metasurface can be relatively constant over a range of wavelengths.This constant phase function, however, is not sufficient to obtain anachromatic behavior.

SUMMARY

Achromatic metasurface optical devices and methods for dispersive phasecompensation using achromatic metasurface optical components aredescribed herein, where various embodiments of the apparatus and methodsfor their fabrication and use may include some or all of the elements,features and steps described below.

An embodiment of an achromatic metasurface optical device includes asubstrate including a surface and a pattern of dielectric resonators onthe surface of the substrate, wherein the dielectric resonators havenonperiodic gap distances between adjacent dielectric resonators; andeach dielectric resonator having a width, w, that is distinct from thewidth of other dielectric resonators.

The widths and the gaps of the dielectric resonators can be configuredto deflect a plurality of wavelengths of interest to or from a focalpoint at a shared focal length. In other embodiments, the widths and thegaps of the dielectric resonators can be configured to deflect aplurality of wavelengths of interest at a shared angle. In additionalembodiments, the widths and gaps of the dielectric resonators can beconfigured to form a same complex wave-front (such as a vortex beam or aBessel beam for a plurality of wavelengths of interest). In particularembodiments, the resonators can have a rectangular cross-section in aplane perpendicular to the substrate surface.

In a method for dispersive phase compensation using achromaticmetasurface optical components, multi-wavelength light is directed to anoptic including a substrate and achromatic metasurface opticalcomponents deposited on a surface of the substrate, wherein theachromatic metasurface optical components comprise a pattern ofdielectric resonators, the dielectric resonators having nonperiodic gapdistances between adjacent dielectric resonators; and each dielectricresonator having a width, w, that is distinct from the width of otherdielectric resonators. A plurality of wavelengths of interest selectedfrom the wavelengths of the multi-wavelength light are deflected withthe achromatic metasurface optical components at a shared angle or to orfrom a focal point at a shared focal length.

The wavelengths of interest can span a range of more than 100 nm.

In particularly embodiments, the substrate comprises silica. Inadditional embodiments, the dielectric resonators comprise silicon.

Each of the dielectric resonators can have a width and thickness thatare smaller than the wavelengths of light. Widths of differentdielectric resonators can differ by at least 25 nm. Additionally, eachof the dielectric resonators can have a width of at least 100 nm.

The dielectric resonators can have multiple electric and magneticresonances that overlap at the wavelengths of interest.

In particular embodiments, the surface of the substrate on which theachromatic metasurface optical components are deposited and a surface onan opposite side of the substrate are both flat.

In additional embodiments, light at wavelengths other than thewavelengths of interest (a) is not deflected or (b) is deflected atangles other than the shared angle or is deflected at angles other thanto/from the focal point at the shared focal length. In still furtherembodiments, a majority of the light at wavelengths other than thewavelengths of interest is removed by the optic to provide multibandoptical filtering of the light.

The replacement of bulk refractive elements with flat ones enables theminiaturization of optical components required for integrated opticalsystems. This process comes with the limitation that planar opticssuffer from large chromatic aberrations due to the dispersion of thephase accumulated by light (in the visible or non-visible spectrum)during propagation. We show that this limitation can be overcome bycompensating the dispersion of the propagation phase with thewavelength-dependent phase shift imparted by a metasurface. Wedemonstrate dispersion-free, multi-wavelength dielectric metasurfacedeflectors in the near-infrared and design an achromatic flat lens inthe same spectral region. This design is based on low-loss coupleddielectric resonators that introduce a dense spectrum of modes to enabledispersive phase compensation. Achromatic metasurfaces can be used inapplications, such as multi-band-pass filters, lightweight collimators,and chromatically-corrected imaging lenses.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 includes a series of images providing a comparison betweenrefractive optics 11 (assuming a material with normal dispersion) in Aand B, diffractive optics 13 (C and D), and achromatic metasurfaces 15(E and F), where the deflection or focusing of different wavelengths oflight 22, 24, and 26 are shown in each image. In the first two cases(A-D), the angle of deflection, θ, and the focal length, f; change as afunction of wavelength. The achromatic metasurface (E) and (F),consisting of subwavelength spaced resonators, is designed to preserveits operation (i.e., same θ and f) for multiple wavelengths. In order toachieve this result, the phase shifts, φ_(m,i) and φ_(m,j), imparted bythe metasurface at points, r_(i) and r_(j), of the interface, aredesigned so that the paths, I_(i)=I(r_(i)) and I_(j)=I(r_(j)) areoptically equivalent at different wavelengths.

FIG. 2 illustrates the scattering properties of an isolated siliconrectangular dielectric resonator 18 with dimensions, w=t=350 nm (withinfinite length along the y-axis), excited by a plane wave traveling atnormal incidence along the z-axis. Chart A plots the scatteringefficiency, Q_(scat), which is defined as the ratio of thetwo-dimensional scattering cross-section, which has the dimension of alength, and the geometric length, w, for transverse-magnetic (TM)excitation 12, electric dipole excitation 14 and magnetic dipoleexcitation 16. The grey arrows indicate the resonant frequenciescalculated with the analytical model for the first two modes (TM₁₁ andTM₂₁). Images B and C show the electric field intensity distribution atthe two resonant frequencies obtained with plane wave excitation. Thewhite lines give a schematic representation of the instantaneouselectric field lines around the resonator.

FIG. 3 includes a side view of the metasurface 15 designed for beamdeflection, wherein a 240-μm long array of silicon rectangulardielectric resonators 18 is patterned on a fused silica substrate 20.The effect of the phase profile, φ_(m), is to deflect normally incidenttransverse-electric (TE) polarized light to an angle, θ₀=17° for λ₁=1300nm, λ₂=1550 nm, and λ₃=1800 nm. The metasurface 15 is divided into 240unit cells similar to the one shown in the inset.

FIG. 4 plots the scattering efficiency for one unit cell of themetasurface of FIG. 3 with geometry, s=1 μm, t=400 nm, w₁=300 nm, w₂=100nm, and g=175 nm. The spectrum shows resonances due to the individualelements and to the coupling between the resonators, as shown by theelectric field intensity distributions.

FIG. 5 is a vector representation of the interference between theelectric fields scattered by the slot and by the two resonators,proportional to a and b, respectively. The phase of b associated withthe resonant response can span the range (π/2, 3π/2), as indicated bythe double-line. The vector sum of a (in green) and b is represented bythe phasor, E (orange), for two different wavelengths (solid and dashedlines).

FIG. 6 plots the normalized intensity (solid line) 22 and phase (dashedline) 24 calculated at a distance of 10 cm away on the vertical axis tothe interface for the same unit cell. The crosses represent the requiredphase values calculated from Equation 2 for λ₁, λ₂, and λ₃. The circlescorrespond to the scattered intensities for the same wavelengths.

FIG. 7 is an SEM image of the cross section of the metasurface.

FIG. 8 is a photographic image of a 240 μm×240 μm section of thefabricated metasurface of FIG. 7 taken with an optical microscope.

FIG. 9 is a schematic illustration of the experimental setup, includinga tunable laser source 26 (that produces the incident light 28) and anInGasAs detector 30.

FIG. 10 plots the simulated far-field intensity as a function of theangle, θ, from the normal to the interface for λ₁=1300 nm 32, λ₂=1550 nm34, and λ₃=1800 nm 36.

FIG. 11 plots the measured far-field intensity as a function of theangle, θ, from the normal to the interface. The intensity is normalizedto the maximum value for the three wavelengths. The inset plot in FIG.11 is close-up around the angle, θ₀.

FIG. 12 plots experimentally measured deflection angles (circles) andsimulated deflection angles (squares) for wavelengths from 1100 nm to1950 nm. The curves correspond to the predicted deflection anglecalculated from Equation 2 for fixed phase gradients designed forθ₀=−17° and λ=1300 nm 32, 1550 nm 34, and 1800 nm 36, respectively.

FIG. 13 plots the intensity measured by the detector at 00; the threepeaks at the wavelengths, λ₁, λ₂ and λ₃, have similar intensities and ahigh suppression ratio (50:1) with respect to other wavelengths.

FIG. 14 shows the results of a simulation of an achromatic flat lens 38based on rectangular dielectric resonators. Illustration A shows abroadband plane wave 28 illuminating the backside of the cylindricallens 38 with side, D=600 μm, and focal distance, f=7.5 mm. Images B-Hshow the far-field intensity distribution for different wavelengths. Thedashed lines correspond to the desired focal planes.

FIG. 15 plots the cross section across the focal plane of the intensitydistribution for λ₁, λ₂, and λ₃ for the achromatic flat lens 38 of FIG.14 with rectangular dielectric resonators.

FIG. 16 plots the focal lengths as a function of wavelength, calculatedas the distance between the lens center and highest intensity point onthe optical axis, for the achromatic flat lens of FIG. 14 withrectangular dielectric resonators. The three larger markers 40correspond to the wavelengths of interest.

FIG. 17 plots an ellipsometric characterization of the 400-nm-thin a-Sifilm deposited with PECVD. The thinner curve 42 is obtained by fittingthe experimental data 44 with the analytical model. The imaginary partof the refractive index is negligible at the wavelengths of interest(1100 nm-2000 nm).

FIG. 18 shows the geometry and field distribution of a rectangulardielectric resonator 18.

FIG. 19 charts the scattering cross section for a silicon rectangulardielectric resonator in vacuum with geometry, w=400 nm and t=500 nm,excited with TM polarization.

FIG. 20 charts a comparison between the theoretical model 46 and a FDTDsimulation 48 of the resonant wavelengths for the first three modes(TM₁₁, TM₁₂, TM₁₃).

FIGS. 21 and 22 provide a comparison between the theoretical model andFDTD simulations of the resonant wavelengths for TE excitation fordifferent widths, w, and for t=300 nm (FIG. 21 ) and t=500 nm (FIG. 22).

FIGS. 23 and 24 provide a comparison between the theoretical model andFDTD simulations of the resonant wavelengths for TM excitation fordifferent widths, w, and for t=300 nm (FIG. 23 ) and t=500 nm (FIG. 24).

FIG. 25 plots the far-field measurement of the beam deflector for A₁, A₂and A₃. Since the structure does not present any periodicity, there isno peak in correspondence of the −1 order (0=17°).

FIGS. 26-29 plot a FDTD simulation of the beam deflector performance fornon-normal incidence, where the incoming beam forms an angle of −1°(FIG. 26 ), +1° (FIG. 27 ), −3° (FIG. 28 ) and +8° (FIG. 29 ) withrespect to the normal. The orange arrows indicate the expecteddeflection angles for an achromatic metasurface.

FIG. 30 plots the performance of the beam deflector as a multi-bandfilter, where a FDTD simulation confirms the uniformity and suppressionratio of the experimental data. The inset shows a close up of the peakcorresponding to A₂ from which we can estimate the FWHM bandwidth of thefilter.

In the accompanying drawings, like reference characters refer to thesame or similar parts throughout the different views. The drawings arenot necessarily to scale; instead, emphasis is placed upon illustratingparticular principles in the exemplifications discussed below.

DETAILED DESCRIPTION

The foregoing and other features and advantages of various aspects ofthe invention(s) will be apparent from the following, more-particulardescription of various concepts and specific embodiments within thebroader bounds of the invention(s). Various aspects of the subjectmatter introduced above and discussed in greater detail below may beimplemented in any of numerous ways, as the subject matter is notlimited to any particular manner of implementation. Examples of specificimplementations and applications are provided primarily for illustrativepurposes.

Unless otherwise herein defined, used or characterized, terms that areused herein (including technical and scientific terms) are to beinterpreted as having a meaning that is consistent with their acceptedmeaning in the context of the relevant art and are not to be interpretedin an idealized or overly formal sense unless expressly so definedherein. For example, if a particular composition is referenced, thecomposition may be substantially, though not perfectly pure, aspractical and imperfect realities may apply; e.g., the potentialpresence of at least trace impurities (e.g., at less than 1 or 2%) canbe understood as being within the scope of the description; likewise, ifa particular shape is referenced, the shape is intended to includeimperfect variations from ideal shapes, e.g., due to manufacturingtolerances. Percentages or concentrations expressed herein can representeither by weight or by volume. Processes, procedures and phenomenadescribed below can occur at ambient pressure (e.g., about 50-120kPa—for example, about 90-110 kPa) and temperature (e.g., −20 to 50°C.—for example, about 10-35° C.) unless otherwise specified.

Although the terms, first, second, third, etc., may be used herein todescribe various elements, these elements are not to be limited by theseterms. These terms are simply used to distinguish one element fromanother. Thus, a first element, discussed below, could be termed asecond element without departing from the teachings of the exemplaryembodiments.

Spatially relative terms, such as “above,” “below,” “left,” “right,” “infront,” “behind,” and the like, may be used herein for ease ofdescription to describe the relationship of one element to anotherelement, as illustrated in the figures. It will be understood that thespatially relative terms, as well as the illustrated configurations, areintended to encompass different orientations of the apparatus in use oroperation in addition to the orientations described herein and depictedin the figures. For example, if the apparatus in the figures is turnedover, elements described as “below” or “beneath” other elements orfeatures would then be oriented “above” the other elements or features.Thus, the exemplary term, “above,” may encompass both an orientation ofabove and below. The apparatus may be otherwise oriented (e.g., rotated90 degrees or at other orientations) and the spatially relativedescriptors used herein interpreted accordingly.

Further still, in this disclosure, when an element is referred to asbeing “on,” “connected to,” “coupled to,” “in contact with,” etc.,another element, it may be directly on, connected to, coupled to, or incontact with the other element or intervening elements may be presentunless otherwise specified.

The terminology used herein is for the purpose of describing particularembodiments and is not intended to be limiting of exemplary embodiments.As used herein, singular forms, such as “a” and “an,” are intended toinclude the plural forms as well, unless the context indicatesotherwise. Additionally, the terms, “includes,” “including,” “comprises”and “comprising,” specify the presence of the stated elements or stepsbut do not preclude the presence or addition of one or more otherelements or steps.

Additionally, the various components identified herein can be providedin an assembled and finished form; or some or all of the components canbe packaged together and marketed as a kit with instructions (e.g., inwritten, video or audio form) for assembly and/or modification by acustomer to produce a finished product.

Dispersive Phase Compensation

A desired optical functionality (e.g., focusing, beaming, etc.) requiresconstructive interference between multiple light paths separating theinterface and the desired wavefront (i.e., the same total accumulatedphase, ϕ_(tot), modulo 2π for all light paths, as shown in images E andF of FIG. 1 ). The total accumulated phase is the sum of the followingtwo contributions: φ_(tot)(r, λ)=φ_(m)(r, λ)+φ_(p) (r, λ), where φ_(m)is the phase imparted at point, r, by the metasurface 15, where φ_(p) isthe phase accumulated via propagation through free space, and where λ isthe wavelength of light. The first term is related to the scattering ofthe individual metasurface elements and is characterized by asignificant variation across the resonance. The second term is given by

${{\varphi_{p}\left( {r,\lambda} \right)} = {\frac{2\pi}{\lambda}1\left( {r,\lambda} \right)}},$where I(r) is the physical distance between the interface at position,r, and the desired wavefront (as shown in images E and F of FIG. 1 ). Toensure achromatic behavior of the device (e.g., with deflection angle orfocal length independent of wavelength), the condition of constructiveinterference should be preserved at different wavelengths by keepingφ_(tot) constant. The dispersion of φ_(m) is designed to compensate forthe wavelength-dependence of φ_(p) via the following equation:

$\begin{matrix}{{{\varphi_{m}\left( {r,\lambda} \right)} = {{- \frac{2\pi}{\lambda}}1(r)}},} & (1)\end{matrix}$where I(r) contains information on the device function {i.e., beamdeflector [N. Yu, et al, “Flat optics: Controlling wavefronts withoptical antenna metasurfaces,” IEEE J. Sel. Top. Quantum Electron.19(3), 4700423 (May 2013) and F. Aieta, et al, “Out-of-plane reflectionand refraction of light by anisotropic optical antenna metasurfaces withphase discontinuities,” 12 Nano Lett. 1702-1706 (27 Feb. 2012)], lens,axicon [F. Aieta, “Aberration-free ultrathin flat lenses and axicons attelecom wavelengths based on plasmonic metasurfaces,” 12 Nano Lett.4932-36 (21 Aug. 2012)], etc.}. Equation 1 is the cornerstone for thedesign of an achromatic metasurface 15. This approach to flat opticsfeatures the advantages of diffractive optics 13, such as flatness andsmall footprint, while achieving achromatic operation. As an example ofan achromatic metasurface 15, we demonstrate a dispersion-free beamdeflector based on dielectric resonators 18. While the typical functionof a diffractive grating is the angular separation of differentwavelengths, we show beam deflection with a wavelength-independent angleof deflection, θ, for a discrete set of wavelengths (λ₁=1300 nm, λ₂=1550nm, and λ₃=1800 nm).

The basic unit of the achromatic metasurface 15 is a resonator 18 thatcan be designed to adjust the scattered phase at different wavelengths,φ_(m)(r, λ), in order to satisfy Equation 1. In particular embodiments,the resonators 18 are dielectric antennas (i.e., resonant elements thatinteract with electromagnetic waves via a displacement current and thatcan have both electric and magnetic resonances). Primarily used in themicrowave frequency range, dielectric antennas have recently beenproposed in the optical regime as an alternative to metallic antennasbecause of their low losses at shorter wavelengths. Nanostructures madeof a material with a large refractive index exhibit resonances whileremaining small compared to the wavelength of light in free-space,similar to what occurs in plasmonic antennas.

Design of Dielectric Achromatic Metasurfaces

To design an achromatic metasurface 15, the scattering properties of arectangular dielectric resonator (RDR), which is a resonator 18 withrectangular cross-section in the x-z plane and infinite extent along they axis, were studied, as shown in the inset of FIG. 2 . Despite thesimple geometry, an analytical closed-form for the electromagneticfields does not exist for rectangular dielectric resonators; therefore,designs described herein are optimized using finite-differencetime-domain (FDTD) simulations. However, in order to estimate thespectral position of the resonant modes, an approximated solution basedon the dielectric waveguide model is derived. The model predicts theexistence of a transverse magnetic (TM_(mn)) mode 12 and a transverseelectric (TE_(mn)) mode inside the resonator. TM modes 12 are excited byan electric field with a polarization parallel to the side, w, of therectangular dielectric resonator, while TE modes are activated by anexcitation polarized along the y-axis. The subscripts, m and n, denotethe number of field extrema in the x- and z-directions. The derivationof the model and a detailed comparison with FDTD simulations arereported in the Exemplification section, below.

Plot A of FIG. 2 shows scattering efficiencies calculated from FDTDsimulations for an isolated silicon rectangular dielectric resonator invacuum with geometry, w=t=350 nm, and excited with TM-polarized light(black line). Analogous to the scattering of dielectric spheresrigorously described by Mie theory, the first two peaks correspond tothe electric and magnetic dipole resonances of the electric dipoleexcitation 14 and the magnetic dipole excitation 16, respectively. Thiscorrespondence is confirmed by showing the scattering spectra of thesame rectangular dielectric resonator independently excited with anelectric and a magnetic dipole placed at the center of the resonator andoriented along the x and y axis, respectively. The grey arrows indicatethe resonant frequencies calculated with the analytical model for thefirst two modes (TM₁₁ and TM₂₁). The electric field intensitydistributions at the two resonances (images B and C of FIG. 2 ) confirmthe electric and magnetic dipole-like scattering. At shorterwavelengths, many higher orders exist with multi-pole-like scattering.

By placing two rectangular dielectric resonators in close proximity suchthat their near fields overlap, a system of coupled resonators 18 iscreated that significantly changes the spectral positions and widths ofthe resonances. We can thus utilize the gap size and position asadditional degrees of freedom to engineer the scattering amplitude andphase. Because of the lack of an analytical solution for coupledrectangular dielectric resonators, we rely on FDTD simulations topredict their optical response.

An achromatic metasurface 15 can be designed by judiciously selecting anappropriate distribution of rectangular dielectric resonators. FIG. 3shows the side view of the metasurface, wherein a 240 μm-long collectionof silicon (Si) resonators patterned on a fused silica (SiO₂) substrateis designed to deflect normally incident light at an angle, θ=−17°, forthree different wavelengths (i.e., λ₁=1300 nm, λ₂=1550 nm, and λ₃=1800nm). The target wavelengths and spatially varying phase functions,represented by the three lines 32, 34, and 36, respectively for λ₁, λ₂,and λ₃ in FIG. 3 are defined by the following equations:

$\begin{matrix}{{{\varphi_{m}\left( {x,\lambda_{i}} \right)} = {{- \frac{2\pi}{\lambda_{i}}}\sin\;\theta_{0}x}},{{{for}\mspace{14mu} I} = 1},2,3.} & (2)\end{matrix}$

We divide the metasurface into 240 slots with width, s=1 μm; and foreach of them, we choose two rectangular dielectric resonators of fixedheight, t=400 nm, and varying widths and separation, w₁, w₂ and g (asshown in FIG. 3 ), so that the phase response follows Equation 2. Eachunit cell, made of a slot and two rectangular dielectric resonators, isdifferent from each of the others; and, therefore, the metasurface 15 iscompletely aperiodic, unlike other blazed gradient metasurfaces.

To demonstrate the mechanism of light control at different wavelengths,consider the phase response required at one particular position of thebeam deflector. From Equation 2, the target phase values for the unitcell centered at the position, x=64 μm for λ₁, λ₂ and λ₃, are calculatedto be φ_(m) ¹=142°, φ_(m) ²=25°, and φ_(m) ³=141°. FIG. 4 shows thescattering cross section of an isolated unit cell with geometry, w₁=300nm, w₂=100 nm, and g=175 nm, excited with TE polarization. Considering aplane wave travelling along the z-axis and incident on the unit cell ata large distance from the interface (i.e.,

>>λ), the field distribution is given by the following twocontributions: the light diffracted by the subwavelength slot and thefield scattered by the coupled resonators, as expressed in the followingequation:

$\begin{matrix}{{{E(\rho)} \approx {\frac{e^{jk\rho}}{\rho}\left\lbrack {a + {b(\theta)}} \right\rbrack}},} & (3)\end{matrix}$where a is the diffraction amplitude proportional to the amount ofincident field that does not interact with the resonators and is inphase with the incident light; θ is the angle between

and the z axis; and b(θ) is the complex scattering function.

Equation 3 is valid in the limit of slot size, s, being significantlysmaller than free-space wavelength, λ, which is not entirely applicablefor our feature size; however, this approximation is sufficient todemonstrate the concept. The interference described by Equation 3 makesit possible to independently control the phase in the 0-2π range atseveral wavelengths simultaneously. This effect can be visualized usingthe complex field (phasors) representation of FIG. 5 . While a is inphase with the incident field, the phase of b, associated with thescattered light due to the TE and TM resonances of the dielectricresonators, spans the range (π/2, 3π/2). The vector sum, E, can thuscover all four quadrants. Note that the scattering cross section,Q_(scat), in FIG. 4 used to visualize the resonance of the structure isrelated to the forward scattering amplitude, b(0), by the opticaltheorem.

FIG. 6 plots the normalized intensity (solid line) 22 and phase (dashedline) 24 calculated at a distance of 10 cm away on the vertical axis tothe interface for the same unit cell and confirms that the coupledrectangular dielectric resonators shown in this example give a fieldwith uniform transmitted intensity, |E|², at the three wavelengths ofinterest (i.e., the circles in FIG. 6 ) and phases, φ_(m) ¹, φ_(m) ²,and φ_(m) ³, matching our design (i.e., the crosses in FIG. 6 ). Whendifferent unit cells composed of the two coupled resonators are placedclose to each other, we expect the mutual coupling between neighboringresonators to partially modify the amplitude and phase response comparedto the isolated cell. However, this interaction does not significantlycompromise the overall response of the structure, as is shown below.

Experimental Realization of a Dispersion-Free Beam Deflector

FDTD simulations were performed to optimize the parameters, w₁, w₂, andg, for each unit cell in order to obtain the desired phase response,φ_(m)(x,λ), and a roughly uniform transmitted amplitude. We fix the unitcell width at s=1 μm, the height of the resonators 18 at t=400 nm, andthe minimum value for w and g at 100 nm to keep the aspect ratio of thestructure compatible with the fabrication process. The algorithmutilized for the selection of the parameters of each unit cell isdescribed in the Exemplification section, below.

The fabrication procedure of the achromatic metasurface can involvechemical vapor deposition of amorphous silicon, electron-beamlithography, and reactive ion etching and is further described in theExemplification section, below. FIGS. 7 and 8 , respectively, show ascanning-electron-microscope (SEM) micrograph of several unit cells andan optical image of the entire fabricated metasurface. The experimentalsetup, represented in FIG. 9 and described in detail in theExemplification section, below, allows measurement (by a detector 30) ofthe far field intensity distribution of the light 28 from the tunablelaser source 26 after the light 28 is scattered by the metasurface 15 inthe A range from 1100 to 2000 nm. From the FDTD simulation of the entirestructure, we calculate the far-field distribution of light transmittedthrough the interface at several wavelengths. Both the simulation (shownin FIG. 10 ) and the experimental results (FIG. 11 ), which plots thetransmitted intensity across a range of angles for λ₁=1300 nm 32,λ₂=1550 nm 34 and λ₃=1800 nm 36, show the achromatic behavior of themetasurface 15. While the dispersive nature of any conventionalflat/diffractive optical component would produce an angular separationof the three wavelengths, the angle of deflection at λ₁=1300 nm 32,λ₂=1550 nm 34, and λ₃=1800 nm 36 is the same, θ=−17°. The diffractionorder at the opposite side (−θ₀) is completely suppressed (seeExemplification section, below) confirming that the structure does notpresent any periodicity and that the steering effect is the result ofthe phase gradient introduced by the subwavelength resonators.

FIG. 12 summarizes the deflection angles for normal incidence simulatedand measured in the entire spectral range from 1150 to 1950 nm. Asexpected, the device deflects the incident light at angle, θ₀, only forthe designed wavelengths. The three lines 32, 34, and 36 in FIG. 12 arethe theoretical dispersion curves obtained from Equation 2 formetasurfaces designed for fixed wavelengths, λ₁, λ₂ and λ₃,respectively. The overlap of the experimental and simulated data withthese curves indicates that wavelengths other than λ₁, λ₂ and λ₃ tend tofollow the dispersion curve of the closest designed wavelengths. Thisresult suggests that increasing the number of chromatically correctedwavelengths within a particular bandwidth is a viable path toward thecreation of a truly broadband achromatic metasurface that operates overa continuous wavelength range.

An advantageous objective for an achromatic optical device is uniformefficiency within the bandwidth (1). The intensity at the angularposition, θ=−17°, is measured as a function of the wavelength from 1100nm to 2000 nm in FIG. 13 . This result shows good uniformity of theintensity measured at λ₁, λ₂ and λ₃ (i.e., intensity variations are lessthan 13%) and large suppression ratios with respect to the otherwavelengths (50:1). These properties suggest that this device can beused as an optical filter with multiple pass bands; the full-width athalf-maximum for each band is about 30 nm (more details in theExemplification section, below). Compared to conventional bandpassoptical filters that often rely on thin film interference effects frommulti-layer stacks, a filter based on the achromatic metasurfacesdescribed herein is much thinner and can be created in a single step ofdeposition, lithography, and etching.

The absolute efficiency of the device (total power at θ₀ divided by theincident power) is also measured for the three wavelengths, which is9.8%, 10.3% and 12.6% for λ₁, λ₂ and λ₃, respectively. From the analysisof the FDTD simulations, one can understand the origin of the limitedefficiency and how to improve it. Optical losses are negligible, asexpected, given the low absorption coefficient of silicon (Si) in thenear infrared. For the three wavelengths of interest, the averagetransmitted power is about 40% of the incident power, while theremaining 60% is reflected. The transmitted power that is not directedto the desired angle of deflection goes into residual diffraction orders(note, for example, the intensity peak at θ=0° for λ=1100 nm in FIG. 11). This residual diffraction is mainly due to the imperfect realizationof the phase function and non-uniform resonators' scattering amplitudesacross the metasurface. We expect that a more advanced algorithm for theselection of the resonators' geometry {e.g., genetic algorithms [see D.E. Goldberg, Genetic Algorithms in Search, Optimization, and MachineLearning (Addison-Wesley, 1989)], particle swarm optimization [N. Jin,et al., “Advances in Particle Swarm Optimization for Antenna Designs:Real-Number, Binary, Single-Objective and MultiobjectiveImplementations,” 55 IEEE Trans. Antenn. Propag. 556-557 (March 2007)],etc.}; optimization of the other parameters, s and t; or choice of adifferent type of resonator would yield a more accurate approximation ofthe target phase function, which could bring the efficiency of thedevice up to 40%. The large reflected component is a result of thestrong directionality of the rectangular dielectric resonator scatteringtowards the half-plane with a higher refractive index.

Using a low-index substrate 20 (e.g., porous silica or even an aerogel)would, therefore, increase the efficiency to almost 50%. Recently, astack of three metasurfaces has been proposed to provide complete phasecontrol and to eliminate the reflected power, leading to 100%transmission at a single wavelength. An alternative approach is based onthe control of the spectral position of electric and magnetic dipoleresonances in dielectric resonators 18 to achieve impedance matching. Ithas indeed been shown that when these two resonances have the exact samecontribution to the scattering of a nanoparticle, the interference ofthe two scattering channels with the excitation produces perfecttransmission and zero reflection. The multi-polar resonances observed inthe rectangular dielectric resonators can be separated in electric- andmagnetic-type of resonances depending on the distribution of the fieldsand the scattering properties (as shown in images B and C of FIG. 2 ).By designing dielectric resonators with multiple electric and magneticresonances that overlap at the wavelengths of interests, multi-spectralcontrol of the wavefront with high transmitted power can be achieved.

Note that, in general, the phase function is defined up to an arbitraryadditive constant; therefore, Equation 1 can be generalized as follows:

$\begin{matrix}{{\varphi_{m}\left( {r,\lambda} \right)} = {{{- \frac{2\pi}{\lambda}}1(r)} + {{C(\lambda)}.}}} & (4)\end{matrix}$

For linear optics applications, C(λ) can take on any value and thus canbe used as a free parameter in the optimization of the metasurfaceelements. More generally, C(λ) can be an important design variable inthe regime of nonlinear optics where the interaction between light ofdifferent wavelengths becomes significant.

Achromatic Flat Lens

As a final demonstration of achromatic metasurfaces, a flat lens designbased on rectangular dielectric resonators for the same threewavelengths is presented. The same parameters, s and t, are used as wereused in the previous demonstration; and the values of w₁, w₂ and g for600 unit cells are chosen using a similar optimization code, where thetarget wavelength and spatially variant phase function is expressed asfollows:

$\begin{matrix}{{{\varphi_{m}\left( {x,\lambda_{i}} \right)} = {{{- \frac{2\pi}{\lambda_{i}}}\left( {\sqrt{x^{2} + f^{2}} - f} \right){\mspace{11mu}\;}{for}\mspace{14mu} i} = 1}},2,3,} & (5)\end{matrix}$where the focal distance is f=7.5 mm.

Since two-dimensional rectangular dielectric resonators are being used,the hyperbolic phase gradient is applied only in one dimension,imitating a cylindrical lens. The achromatic properties of the lens aredemonstrated with FDTD simulations, as shown in FIG. 14 . Broadbandlight 28 from a light source (e.g., a laser, a light bulb, or the sunthat transmits light through a polarizer that transmits light polarizedalong the axis of the rectangular dielectric resonator) illuminates thebackside of the flat lens 38 at normal incidence (as shown in image A ofFIG. 14 ). The intensity distribution at different wavelengths shows theexpected focusing at f=7.5 mm for λ₁, λ₂, and λ₃ (as shown in images C,E, and G of FIG. 14 ) and aberrated focusing at other wavelengths (asshown in images B, D, F, and H of FIG. 6 ). The diameters of the Airydisks at the focal spots are 50, 66 and 59 μm for λ₁, λ₂, and λ₃,respectively, achieving focusing close to the diffraction limit (40 μm,47 μm, and 55 μm, for numerical aperture, NA=0.05) (as shown in image Iof FIG. 6 ). For the wavelengths close to λ₁, λ₂ and λ₃, the focaldistance follows the dispersion curve associated with the closestcontrolled wavelength (as shown in as shown in image J of FIG. 6 ). Wenote that a recent report [C. Saeidia, et al., “Wideband plasmonicfocusing metasurfaces,” Appl. Phys. Lett. 105, 053107 (2014)] pointedout that, in order to achieve broadband focusing, the phase shiftdistribution of a metasurface should satisfy a wavelength-dependentfunction, though a general approach to overcome this inherent dispersiveeffect was not provided therein.

Concluding Remarks

Metasurfaces 15 have significant potential as flat, thin and lightweightoptical components that can combine several functionalities into asingle device, making metasurfaces good candidates to augmentconventional refractive or diffractive optics. The achromaticmetasurface concept demonstrated here can solve one of the most criticallimitations of flat optics (i.e., single wavelength operation).

After introducing the basic concept of dispersion-compensated phase, aplanar beam deflector was demonstrated that is capable of steering lightto the same direction at three different wavelengths and which can alsobe used as a single-layer multi-pass-band optical filter. Additionally,a design was presented for an achromatic flat lens 38 using the samemetasurface 15 building blocks. In the visible realm, this kind of lenscan find application in digital cameras where a red-green-blue (RGB)filter is used to create a color image. Holographic 3D displays requirean RGB coherent wavefront to reconstruct a 3D scene. The use ofachromatic flat optics for the collimation of the backlight may helpmaintain the flatness of such screens. Achromatic metasurfaces 15 forseveral discrete wavelengths can also be implemented in compact andintegrated devices for second harmonic generation, four wave mixing [C.Jin, “Waveforms for Optimal Sub-keV High-Order Harmonics withSynthesized Two- or Three-Color Laser Fields,” et al., 5 Nat. Comm. 4003(30 May 2014)], and other nonlinear processes. The metasurface designdescribed herein is scalable from the ultraviolet (UV) to the terahertz(THz) and beyond, and can be realized with conventional fabricationapproaches (e.g., one step each of deposition, lithography, andetching). Finally, the versatility in the choice of thewavelength-dependent phase allows for functionalities that are verydifferent (even opposite) from achromatic behavior. For example, anoptical device with enhanced dispersion (e.g., a grating able toseparate different colors further apart) can be useful for ultra-compactspectrometers.

Exemplification

Fabrication of Achromatic Metasurfaces

A device was fabricated by depositing 400-nm amorphous silicon (a-Si) ona fused silica (SiO₂) substrate at 300° C. by plasma-enhanced chemicalvapor deposition (PECVD). The rectangular dielectric resonators weredefined by electron-beam lithography using the positive resist, ZEP-520Afrom ZEON Corp., diluted in Anisole with a ratio of 1:1; exposed to adose of 300 μC/cm2 (500 pA, 125 kV); and developed for 50 sec at roomtemperature in o-xylene. The silicon ridges were then obtained by dryetching using Bosch processing. At the end of the process, the residualresist layer was removed with a one-hour bath in MICROPOSIT Remover 1165(from Rohm and Haas Electronic Material, LLC, of Marlborough, Mass.,US), rinsed in PG Remover (from MicroChem Corp. of Newton, Mass., US)and exposed to 1 minute of O₂ plasma at 75 W. The sample used for theSEM image in FIG. 7 was sputter-coated with 5 nm of platinum/palladiumto eliminate charging in the SEM.

Optical Properties of the Amorphous Silicon

The plot of FIG. 17 shows the experimental data 44 (and model fit 46) ofthe optical properties of the amorphous silicon layer in the wavelengthrange of 400 nm to 850 nm obtained with an Imaging Ellipsometer“nanofilm_ep4” performed by Accurion. The Cody-Lorentz dispersion modelwas used to extrapolate the refractive index into the near infrared. Thevalues extracted were used for the numerical simulations.

Experimental Setup

The measurement set-up includes a supercontinuum laser (e.g., “SuperK”laser from NKT Photonics of Birkerød, Denmark) equipped with a set ofacousto-optic tunable filters (NKT “Select” filters) to tune theemission from 1100 nm to 2000 nm with a line-width of 15 nm. The outputof the laser is focused with a long focal distance lens (f=20 cm, notshown in FIG. 9 ) to guarantee uniform illumination of the 240 μm×240 μmmetasurface. The intensity of the transmitted light as a function of theangle, θ, is recorded by using a broadband InGaAs detector (ThorlabsDET10D) mounted on a motorized rotation stage. In FIG. 12 , theexperimental data below λ=1300 nm are missing because of the low powerof the source and low sensitivity of the detector below that wavelength.For the measurement of the efficiency, the detector is replaced by thehead of a power meter (Ge photodiode sensor) with large active area. Theefficiency values are normalized to the power incident on the back ofthe device.

Algorithm for the Optimization of the Unit-Cells

To implement a given functionality of the achromatic metasurface 15, aparticular wavelength-dependent phase function (Equation 1) is realizedby designing the scattering properties of unit cells consisting ofcoupled dielectric resonators 18.

We fix the unit cell width, s=1 μm; the height of the siliconresonators, t=400 nm; and the minimum value for wand gat 100 nm; and werun a cycle of FDTD simulations for different geometries to obtain thedesired phase response, φ_(m)(x, λ), and quasi-uniform transmittedamplitude. We swept the parameters, w₁, w₂, and g in the range from 100nm to 950 nm with steps of 25 nm in all the possible combinationsenforcing that the sum of w₁, w₂ and g did not exceed the size of theunit cell, s, and calculated the transmitted intensity and the phase ata distance of 10 cm away on the vertical to the interface. The phaseresponse was calculated as the phase of the field at that point minusthe phase accumulated by the light via propagation through the glassslab and the air above the unit cell. For each simulation, if thetransmitted intensity is at least 35% of the total source power and thedifference between the calculated phase at each wavelength and thetarget value for a specific unit cell is less than 60°, the set ofparameters is saved for that specific unit cell. The root-mean-squareerror (RMSE) of the phase for the three wavelengths is also calculatedand saved. Every time a new set of parameters passes the check-test fortransmitted intensity and phase difference for a specific unit cell, thegeometry corresponding to the minimum RMSE is retained.

For the design of the beam deflector demonstrated herein, the averageRMSE of the phase among all the unit cells for the three wavelengths atthe end of the optimization is about 30°. This causes an imperfect matchwith the design requirements that will somewhat reduce the performanceof the device (i.e., residual diffraction orders and background).

Rectangular Dielectric Resonator Model

A simple analytical expression based on a dielectric waveguide model(DWM) is derived to estimate the resonant frequencies of a rectangulardielectric resonator. According to this model, an isolated rectangulardielectric resonator is assumed to be a truncated section of an infinitedielectric waveguide, and the field pattern inside the resonator 18 is astanding wave along the x axis inside the dielectrics and decaysexponentially outside (as shown in FIG. 18 ). If we truncate along the zaxis, a standing wave pattern is setup along z, as well. The standingwaves along x and z can be assumed to be governed by the same equations.

After writing the field components and imposing the boundary conditions,we can derive the transcendental equations from which the wave numbers,k_(x) and k_(z), corresponding to the resonant wavelengths can becalculated.

In FIG. 18 , the TM modes are calculated by solving the Helmholtzequation, as follows:∇² H ₁ +k ₀ε_(r) H _(y)=0.

Assuming an harmonic field, the Ampere law provides the following:

${{\nabla \times H} = \left. {ɛ_{r}\frac{\partial E}{\partial t}}\rightarrow\left\{ \begin{matrix}{E_{x} = {\frac{j}{\omega ɛ_{r}}\frac{\partial H_{y}}{\partial z}}} \\{E_{z} = {{- \frac{j}{\omega ɛ_{r}}}\frac{\partial H_{y}}{\partial x}}}\end{matrix} \right. \right.}.$

Given the symmetry of the structure with respect to x=0, the expressionsof the fields inside the resonator, and in the half-planes left (x>w/2),right (x<−w/2), up (z>t/2) and down (z<−t/2) are as follows:

${\text{-}{w/2}} < x < {{\text{-}{w/2}\mspace{14mu}{and}} - {t/2}} < t < {\text{-}{t/2}\text{:~~~}\left\{ {\begin{matrix}{H_{y} = {A\;{\cos\left( {k_{x}x} \right)}{\cos\left( {k_{z}z} \right)}}} \\{E_{x} = {- {\frac{j}{{\omega ɛ}_{r}}Ak_{z}\;{\cos\left( {k_{x}x} \right)}{\sin\left( {k_{z}z} \right)}}}} \\{E_{z} = {\frac{j}{\omega ɛ_{r}}Ak_{x}{\sin\left( {k_{x}x} \right)}{\cos\left( {k_{z}z} \right)}}}\end{matrix},{{z > {t/2}};{z < {\text{-}{t/2}\text{:}\mspace{14mu}\left\{ {\begin{matrix}{H_{y} = {B{\cos\left( {k_{x}x} \right)}e^{- {k_{zo}{({z - {t/2}})}}}}} \\{E_{x} = {{- \frac{j}{\omega}}{Bk}_{zo}{\cos\left( {k_{x}x} \right)}e^{- {k_{zo}{({z - {t/2}})}}}}} \\{E_{z} = {\frac{j}{\omega}{Bk}_{x}{\sin\left( {k_{x}x} \right)}e^{- {k_{zo}{({z - {t/2}})}}}}}\end{matrix},{{{{and}x} > {w/2}};{x < {\text{-}{w/2}\text{:~~~}\left\{ {\begin{matrix}{H_{y} = {Ce^{- {k_{xo}{({x - {w/2}})}}}{\cos\left( {k_{z}x} \right)}}} \\{E_{x} = {{- \frac{j}{\omega}}Ck_{z}e^{- {k_{xo}{({x - {w/2}})}}}{\sin\left( {k_{z}z} \right)}}} \\{E_{z} = {\frac{j}{\omega}Ck_{xo}e^{- {k_{xo}{({x - {w/2}})}}}{\cos\left( {k_{z}x} \right)}}}\end{matrix},} \right.}}}} \right.}}}} \right.}$where A, B, and C are variables to be calculated. The boundaryconditions at the edges of the rectangular dielectric resonator read asfollows:

$\left\{ {\begin{matrix}{E_{z,{IN}} = E_{z,{L/R}}} & {x = {w/2}} \\{E_{x,{IN}} = E_{x,{U/D}}} & {z = {t/2}} \\{{ɛ_{r}E_{x,{IN}}} = E_{x,{L/R}}} & {x = \frac{w}{2}} \\{{ɛ_{r}E_{z,{IN}}} = E_{z,{U/D}}} & {z = {t/2}}\end{matrix}.} \right.$

Applying these conditions, finally, one obtains the following:

$\begin{matrix}{\left\{ \begin{matrix}{{A = {k_{xo}ɛ_{r}}},\ {C = {k_{x}{\sin\left( {k_{x}{w/2}} \right)}}}} \\{B = {\frac{k_{xo}k_{z}}{k_{zo}}{\sin\left( {k_{z}{t/2}} \right)}}} \\{{k_{x}w} = {{m\;\pi} - {2{\tan^{- 1}\left( {k_{x}/\left( {ɛ_{r}k_{xo}} \right)} \right)}}}} \\{{k_{z}w} = {{p\;\pi} - {2{\tan^{- 1}\left( {k_{z}/\left( {ɛ_{r}k_{zo}} \right)} \right)}}}}\end{matrix} \right..} & (6)\end{matrix}$

Using the Following Expressions:

$\left\{ {\begin{matrix}{{k_{x}^{2} + k_{z}^{2}} = {ɛ_{r}k_{0}^{2}}} \\{k_{xo} = \sqrt{{\left( {ɛ_{r} - 1} \right)k_{0}^{2}} - k_{x}^{2}}} \\{k_{zo} = \sqrt{{\left( {ɛ_{r} - 1} \right)k_{0}^{2}} - k_{z}^{2}}}\end{matrix},} \right.$the last two equations of Equation 6 can be solved to give thewavevectors along the x and z axes, corresponding to the resonant modes.

For TE modes, the Helmholtz equation for the electric field is used;and, following a similar procedure, the following transcendentalequations for the resonant wavevectors are obtained:

$\left\{ \begin{matrix}{{k_{x}w} = {{m\;\pi} - {2{\tan^{- 1}\left( {k_{x}/k_{xo}} \right)}}}} \\{{k_{z}w} = {{p\pi} - {2{\tan^{- 1}\left( {k_{z}/k_{zo}} \right)}}}}\end{matrix} \right..$

This model is useful to design a rectangular dielectric resonatorbecause it helps us to predict the spectral positions of the resonantmodes for a given geometry. The predictions of the model were validatedby comparing the results with FDTD simulations. The scattering crosssection of an isolated rectangular dielectric resonator 18 for TMexcitation, such as the one in FIG. 2 with w=400 nm and t=500 nm, issimulated, allowing us to visualize the resonances in terms of thedistribution of the electric and magnetic fields inside the resonator.In the range of wavelengths from 1000 nm to 3400 nm, the followingresonant modes, TM₁₁, TM₁₂ and TM₁₃ are observed (as shown in FIG. 19 ).In FIG. 20 , the resonant wavelengths in the FDTD simulation 48 arecompared with those calculated from the theoretical model 46; theresults are in close agreement, with an error of ±5% consistent withother works in the literature.

We also performed a comprehensive comparison of resonant wavelengths inthe model 46 with resonant wavelengths in simulations 48 by calculatingthe first resonant mode for TE and TM excitation for differentgeometries of the rectangular dielectric resonator. The results arereported in FIGS. 21-24 .

Far Field Measurement at −θ₀

The achromatic beam deflector presented here does not feature anystructural periodicity. While in previous works, a metasurfacefunctionally equivalent to a blazed-grating was designed by repeating asingle unit cell [see, e.g., N. Yu, et al., “Light Propagation withPhase Discontinuities: Generalized Laws of Reflection and Refraction,”334 Science 333-37 (2011); N. Yu, et al., “Flat optics: Controllingwavefronts with optical antenna metasurfaces,” IEEE J. Scl. Top. QuantumElectron. 19(3), 4700423 (2013); N. Yu, et al., “Flat optics withdesigner metasurfaces,” 13 Nat. Materials 139-150 (2014); and F. Aieta,et al., “Out-of-plane reflection and refraction of light by anisotropicoptical antenna metasurfaces with phase discontinuities,” 12 Nano Lett.1702-1706 (27 Feb. 2012)], in the present case, all unit cells aredifferent from each other because the three phase ramps necessary todeflect different wavelengths do not have any periodicity. Therefore, weexpect complete suppression of any residual −1 diffraction order at theangular position, −θ₀, that may rise from the imperfect phase oramplitude profile of the metasurface 15. This suppression is confirmedby looking at the measured far-field intensity distribution for the fullrange of angles, −30° to 30° (FIG. 25 ). Although we see some peaksoutside the desired beaming angle (as discussed in the main text), nointensity peak is measured at 17° (−θ₀), confirming the absence of anyresidual effect due to periodicity.

Angle of Incidence Dependence

The metasurfaces 15 herein described are designed to work with lightarriving at normal incidence. When the incoming beam arrives at anon-orthogonal angle, the symmetry of excitation of the unit cell isbroken. As a consequence, other modes will be excited in thetwo-coupled-resonators system, affecting the phase and amplituderesponse. Under this condition, the device does not perform as anachromatic metasurface 15.

For angles of incidence in the range ±1°, achromatic deflection ispreserved (see FIGS. 26 and 27 ). FIGS. 28 and 29 show the simulatedfar-field intensity distribution of the beam deflector for lightincident from air at −3° and 8° angles with respect to the normal. Inthese cases, the angles of deflection are not constant for the threewavelengths and do not match the angles that we would expect if thephase gradients were those described in Equation 2. While, with thedesign used here, maintaining the desired phase and amplitude responsefor off-normal excitation is difficult, we expect that by reducing thethickness of the rectangular dielectric resonator or by choosing adifferent type of resonator (e.g., a plasmonic antenna), the effect ofoblique illumination will have a smaller impact on the resonantresponse, enabling the achromatic operation for a wide range of anglesof incidence.

Multiband Filter

In the preceding text, we described how a multiband beam deflector canbe used as an optical filter with multiple pass bands. Illuminating themetasurface with broadband light, only the light at λ₁, λ₂ and λ₃ willbe directed to the desired angle, creating a spatial filter. FIG. 30shows the FDTD simulation of the intensity monitored at the angle,θ=−17°, confirming the good intensity uniformity between λ₁, λ₂ and λ₃and a high suppression ratio with respect to the other wavelengths, asseen in the experimental data. We can also estimate the bandwidth of thefilter by looking at the full-width-half-maximum of the three peaks. Asshown in the inset of FIG. 30 for the peak at λ₂, the FWHM isapproximately 30 nm. By designing a more accurate phase function (asdiscussed in the preceding text), the FWHM of the multi-pass band filtercan be reduced.

In describing embodiments of the invention, specific terminology is usedfor the sake of clarity. For the purpose of description, specific termsare intended to at least include technical and functional equivalentsthat operate in a similar manner to accomplish a similar result.Additionally, in some instances where a particular embodiment of theinvention includes a plurality of system elements or method steps, thoseelements or steps may be replaced with a single element or step;likewise, a single element or step may be replaced with a plurality ofelements or steps that serve the same purpose. Further, where parametersfor various properties or other values are specified herein forembodiments of the invention, those parameters or values can be adjustedup or down by 1/100^(th), 1/50^(th), 1/20^(th), 1/10^(th), ⅕^(th),⅓^(rd), ½, ⅔^(rd), ¾^(th), ⅘^(th), 9/10^(th), 19/20^(th), 49/50^(th),99/100^(th), etc. (or up by a factor of 1, 2, 3, 4, 5, 6, 8, 10, 20, 50,100, etc.), or by rounded-off approximations thereof, unless otherwisespecified. Moreover, while this invention has been shown and describedwith references to particular embodiments thereof, those skilled in theart will understand that various substitutions and alterations in formand details may be made therein without departing from the scope of theinvention. Further still, other aspects, functions and advantages arealso within the scope of the invention; and all embodiments of theinvention need not necessarily achieve all of the advantages or possessall of the characteristics described above. Additionally, steps,elements and features discussed herein in connection with one embodimentcan likewise be used in conjunction with other embodiments. The contentsof references, including reference texts, journal articles, patents,patent applications, etc., cited throughout the text are herebyincorporated by reference in their entirety; and appropriate components,steps, and characterizations from these references may or may not beincluded in embodiments of this invention. Still further, the componentsand steps identified in the Background section are integral to thisdisclosure and can be used in conjunction with or substituted forcomponents and steps described elsewhere in the disclosure within thescope of the invention. In method claims, where stages are recited in aparticular order—with or without sequenced prefacing characters addedfor ease of reference—the stages are not to be interpreted as beingtemporally limited to the order in which they are recited unlessotherwise specified or implied by the terms and phrasing.

What is claimed is:
 1. A method of altering incident light usingmetasurface optical components, the method comprising: providing asubstrate; depositing metasurface optical components on the surface ofthe substrate, wherein the metasurface optical components comprise apattern of silicon dielectric resonators, the silicon dielectricresonators having gap distances between adjacent silicon dielectricresonators, and each silicon dielectric resonator has a width and athickness; and directing incident light to the metasurface opticalcomponents, wherein the gap distances, the widths, and the thicknessesare configured to scatter the incident light and impart a phase shift,ranging at least from 0 to 2π, on an outgoing light.
 2. The method ofclaim 1, wherein the incident light comprises a wavelength of about 850nm to 2000 nm.
 3. The method of claim 2, wherein the incident lightcomprises a wavelength of about 1100 nm to 1950 nm.
 4. The method ofclaim 3, wherein the incident light comprises a wavelength of about 1300nm, 1550 nm, and/or 1800 nm.
 5. The method of claim 4, wherein theincident light comprises a narrow band light including a full width halfmax of about 30 nm.
 6. The method of claim 1, wherein the silicondielectric resonators comprise amorphous silicon.
 7. The method of claim1, wherein each of the dielectric resonators have a width of at least100 nm.
 8. The method of claim 1, wherein the substrate comprisessilica.
 9. The method of claim 1, wherein each dielectric resonator hasa rectangular cross-section in a plane perpendicular to the substratesurface.
 10. The method of claim 1, wherein the wherein the gap betweenthe dielectric resonators comprises a material with a refractive indexlower than the refractive index of the dielectric pillars.
 11. Ametasurface optical device, comprising: a substrate including a surface;and a pattern of silicon dielectric resonators on the surface of thesubstrate, wherein the silicon dielectric resonators have gap distancesbetween adjacent silicon dielectric resonators, and each silicondielectric resonator has a width and a thickness and wherein the gapdistances, the widths, and the thicknesses are configured to scatterincident light and impart a phase shift, ranging at least from 0 to 2π,on an outgoing light.
 12. The metasurface optical device of claim 11,wherein the incident light comprises a wavelength of about 850 nm to2000 nm.
 13. The metasurface optical device of claim 12, wherein theincident light comprises a wavelength of about 1100 nm to 1950 nm. 14.The metasurface optical device of claim 13, wherein the incident lightcomprises a wavelength of about 1300 nm, 1550 nm, or 1800 nm.
 15. Themetasurface optical device of claim 14, wherein the incident lightcomprises a narrow band of light including a full width half max ofabout 30 nm.
 16. The metasurface optical device of claim 11, wherein thesilicon dielectric resonators comprise amorphous silicon.
 17. Themetasurface optical device of claim 11, wherein each of the dielectricresonators have a width of at least 100 nm.
 18. The metasurface opticaldevice of claim 11, wherein the substrate comprises silica.
 19. Themetasurface optical device of claim 11, wherein each dielectricresonator has a rectangular cross-section in a plane perpendicular tothe substrate surface.
 20. The metasurface optical device of claim 11,wherein the gap between the dielectric resonators comprises a materialwith a refractive index lower than the refractive index of thedielectric pillars.